A229290 -id:A229290 - cp's OEIS frontend

A229289 Primes p of the form p = 2^k * m + 1, where (i) m is squarefree and odd, (ii) all primes that divide m are in the sequence, and (iii) k is 0, 1, or 2.

2, 3, 5, 7, 11, 13, 23, 29, 31, 43, 47, 53, 59, 61, 67, 71, 79, 107, 131, 139, 157, 173, 211, 263, 269, 277, 283, 311, 317, 331, 347, 349, 367, 373, 421, 431, 461, 463, 547, 557, 599, 643, 659, 661, 683, 691, 709, 733, 743, 787, 827, 853, 859, 863, 911, 941

Offset: 1

José María Grau Ribas, Oct 05 2013

nonn

Taking m=1 in the definition we get the primes 2, 3, 5.
If n is in A226960, then n is a product of terms of this sequence.
If k is only allowed to be 0 or 1, we get 2, 3, 7, 43 and no more. - Jianing Song, Feb 21 2021
Also prime factors of terms in A341858. It is conjectured that this sequence is infinite. - Jianing Song, Feb 22 2021

Ray Chandler, Table of n, a(n) for n = 1..1000

Cf. A227007, A226960, A007814, A229290, A229291, A289355, A341858.
For the complement, see A289355.
Proper subsequence of A066651.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i, Length[fa[n]]}] ; Os[b_, 1] = True; Os[b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n - 1)/b^IntegerExponent[n - 1, b]] &&IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i, 1]]], {i, Length[fa[n - 1]]}] == {True};G[b_] := Select[Prime[Range[1000]], Os[b, #] &];G[2]

is(n)=if(!isprime(n),return(0)); if(n<13,return(1)); my(k=valuation(n-1,2), m=n>>k, f); if(k>2,return(0)); f=factor(m); if(lcm(f[,2])>1, return(0)); for(i=1,#f~, if(!is(f[i,1]), return(0))); 1 \\ Charles R Greathouse IV, Oct 28 2013

Revised definition from Charles R Greathouse IV, Nov 13 2013

Terms corrected by José María Grau Ribas, Nov 14 2013

A229291 n is in the sequence if n is prime, (n-1)/5^A112765(n-1) is a squarefree number, A112765(n-1) < 3 and every prime divisor of n-1 is in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 11, 23, 31, 43, 47, 67, 71, 139, 151, 211, 283, 311, 331, 431, 463, 659, 683, 691, 863, 907, 947, 967, 1051, 1151, 1291, 1303, 1319, 1367, 1427, 1511, 1699, 1867, 1979, 1987, 2011, 2111, 2131, 2311, 2351, 2531, 3011, 3023, 3083, 3323, 3851, 4099

Offset: 1

José María Grau Ribas, Oct 06 2013

nonn

If n is in A226963 then n is some product of elements of this sequence.

Cf. A112765, A226963, A229289, A229290.

fa = FactorInteger; free[n_] := n == Product[fa[n][[i, 1]], {i,
  Length[fa[n]]}]; Os[b_, 1] = True; Os[b_, 2] = True; Os[ b_, b_] = True; Os[b_, n_] := Os[b, n] = PrimeQ[n] && free[(n - 1)/b^IntegerExponent[n - 1,b]] && IntegerExponent[n - 1, b] < 3 && Union@Table[Os[b, fa[n - 1][[i, 1]]], {i, Length[fa[n - 1]]}] == {True}; G[b_] := Select[Prime [Range[2000]], Os[b, #] &]; G[5]

cp's OEIS Frontend

A229289 Primes p of the form p = 2^k * m + 1, where (i) m is squarefree and odd, (ii) all primes that divide m are in the sequence, and (iii) k is 0, 1, or 2.

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